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A new full-Newton step O(n) infeasible interior-point algorithm for semidefinite optimization

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Abstract

Interior-point methods for semidefinite optimization have been studied intensively, due to their polynomial complexity and practical efficiency. Recently, the second author designed a primal-dual infeasible interior-point algorithm with the currently best iteration bound for linear optimization problems. Since the algorithm uses only full Newton steps, it has the advantage that no line-searches are needed. In this paper we extend the algorithm to semidefinite optimization. The algorithm constructs strictly feasible iterates for a sequence of perturbations of the given problem and its dual problem, close to their central paths. Two types of full-Newton steps are used, feasibility steps and (ordinary) centering steps, respectively. The algorithm starts from strictly feasible iterates of a perturbed pair, on its central path, and feasibility steps find strictly feasible iterates for the next perturbed pair. By using centering steps for the new perturbed pair, we obtain strictly feasible iterates close enough to the central path of the new perturbed pair. The starting point depends on a positive number ζ. The algorithm terminates either by finding an ε-solution or by detecting that the primal-dual problem pair has no optimal solution (X *,y *,S *) with vanishing duality gap such that the eigenvalues of X * and S * do not exceed ζ. The iteration bound coincides with the currently best iteration bound for semidefinite optimization problems.

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References

  1. Alizadeh, F.: Combinatorial optimization with interior point methods and semi-definite matrices. Ph.D. thesis, University of Minnesota, Minneapolis (1991)

  2. Alizadeh, F.: Interior point methods in semidefinite programming with applications to combinatorial optimization. SIAM J. Optim. 5(1), 13–51 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bai, Y.Q., El Ghami, M., Roos, C.: A comparative study of kernel functions for primal-dual interior-point algorithms in linear optimization. SIAM J. Optim. 15(1), 101–128 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bai, Y.Q., Roos, C.: A primal-dual interior-point method based on a new kernel function with linear growth rate. In: Caccetta, L., Rehbock, V. (eds.) Industrial Optimisation. Proceedings of Symposium on Industrial Optimisation and the 9TH Australian Optimisation Day, vol. 1, pp. 15–28. Curtin University of Technology, Australia, Western Australian Centre of Excellence in Industrial Optimisation, Perth (2005)

  5. Ben-Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization, MPS/SIAM Series on Optimization. Analysis, Algorithms, and Engineering Applications. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2001)

    Google Scholar 

  6. Boyd, S.E., El Ghaoui, L., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM Studies in Applied Mathematics. SIAM, Philadelphia, USA (1994)

    Google Scholar 

  7. Boyd, S.E., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

  8. Boyd, S.E., Vandenberghe, L.: Semidefinite programming. SIAM Rev. 38(1), 49–96 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  9. De Klerk, E.: Aspects of Semidefinite Programming. Applied Optimization, vol. 65. Kluwer Academic, Dordrecht (2002)

    Google Scholar 

  10. De Klerk, E., Roos, C., Terlaky, T.: Initialization in semidefinite programming via a self–dual skew-symmetric embedding. Oper. Res. Lett. 20, 213–221 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  11. Elabwabi, G.: Topics in global optimization using semidefinite optimization. Ph.D. thesis, Faculty of Mathematics and Computer Science, TU Delft, NL–2628 CD Delft (2007)

  12. Haeberley, J.-P.A., Alizadeh, F., Overton, M.L.: Complementarity and nondegeneracy in semidefinite programming. Math. Program. 77, 111–128 (1997)

    Google Scholar 

  13. Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. Assoc. Comput. Mach. 42(6), 1115–1145 (1995)

    MATH  MathSciNet  Google Scholar 

  14. Helmberg, C.: Semidefinite programming for combinatorial optimization. Technical Report 00-34, Konrad-Zuse-Zentrum für Informationstechink Berlin, Takustraße 7, D-14195 Berlin (2000)

  15. Helmberg, C., Rendl, F., Vanderbei, R.J., Wolkowicz, H.: An interior point method for semidefinite programming. SIAM J. Optim. 6, 342–361 (1994)

    Article  MathSciNet  Google Scholar 

  16. Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)

    MATH  Google Scholar 

  17. Jansen, B., Roos, C., Terlaky, T., Vial, J.-Ph.: Primal-dual algorithms for linear programming based on the logarithmic barrier method. J. Optim. Theory Appl. 83, 1–26 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  18. Jarre, F.: An interior-point method for minimizing the largest eigenvalue of a linear combination of matrices. SIAM J. Control Optim. 31, 1360–1377 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  19. Karisch, E.S., Rendl, F., Clausen, J.: Solving graph bisection problems with semidefinite programming. INFORMS J. Comput. 12(3), 177–191 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  20. Kojima, M., Megiddo, N., Mizuno, S.: A primal-dual infeasible-interior-point algorithm for linear programming. Math. Program. 61, 263–280 (1993)

    Article  MathSciNet  Google Scholar 

  21. Kojima, M., Mizuno, S., Yoshise, A.: A primal-dual interior point algorithm for linear programming. In: Megiddo, N. (ed.) Progress in Mathematical Programming: Interior Point and Related Methods, pp. 29–47. Springer, New York (1989)

    Google Scholar 

  22. Kojima, M., Shida, M., Shindoh, M.: Local convergence of predictor-corrector infeasible interior-point algorithm for SDPs and SDLCPs. Math. Program. Ser. A 80, 129–160 (1998)

    MathSciNet  Google Scholar 

  23. Kojima, M., Shindoh, M., Hara, S.: Interior point methods for the monotone linear complementarity problem in symmetric matrices. SIAM J. Optim. 7, 86–125 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  24. Lasserre, J.B., Prieto-Rumeau, T.: SDP vs. LP relaxations for the moment approach in some performance evaluation problems. Stoch. Models 20, 439–456 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  25. Lasserre, J.B.: Global optimization with polynomials and the problems of moments. SIAM J. Optim. 11, 796–817 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  26. Lisser, A., Rendl, F.: Graph partitioning using linear and semidefinite programming. Math. Program. Ser. B 95(1), 91–101 (2003); ISMP 2000, Part 3 (Atlanta, GA)

    Article  MATH  MathSciNet  Google Scholar 

  27. Luo, Z.Q., Sturm, J.F., Zhang, S.: Conic convex programming and self-dual embedding. Optim. Methods Softw. 14(3), 196–218 (2000)

    Article  MathSciNet  Google Scholar 

  28. Lustig, I.J.: Feasibile issues in a primal-dual interior point method for linear programming. Math. Program. 49, 145–162 (1990/91)

    Article  MATH  MathSciNet  Google Scholar 

  29. Lütkepohl, H.: Handbook of Matrices. Wiley, Chichester (1996)

    MATH  Google Scholar 

  30. Mansouri, H., Roos, C.: Simplified O(n) infeasible interior-point algorithm for linear optimization using full-Newton step. Optim. Methods Softw. 22(3), 519–530 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  31. Mansouri, H.: Full-newton step interior-point methods for conic optimization. Ph.D. thesis, Faculty of Mathematics and Computer Science, TU Delft, NL–2628 CD Delft (2008)

  32. Megiddo, N.: Pathways to the optimal set in linear programming. In: Megiddo, N. (ed.) Progress in Mathematical Programming: Interior Point and Related Methods, pp. 131–158. Springer, New York (1989)

    Google Scholar 

  33. Mizuno, S.: Polynomiality of infeasible interior point algorithms for linear programming. Math. Program. 67, 109–119 (1994)

    Article  MathSciNet  Google Scholar 

  34. Mizuno, S., Todd, M.J., Ye, Y.: On adaptive-step primal-dual interior-point algorithms for linear programming. Math. Oper. Res. 18, 964–981 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  35. Monteiro, R.D.C., Adler, I.: Interior-path following primal-dual algorithms: part I: Linear programming. Math. Program. 44, 27–41 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  36. Nesterov, Y., Todd, M.J.: Primal-dual interior point methods for self-scaled cones. SIAM J. Optim. 8(2), 324–364 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  37. Nesterov, Y.E., Nemirovski, A.: Interior Point Polynomial Methods in Convex Programming: Theory and Algorithms. SIAM Publications. SIAM, Philadelphia (1993)

    Google Scholar 

  38. Parrilo, P.A.: Semidefinite programming relaxations for semialgebraic problems. Math. Program. 96, 293–320 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  39. Peña, J., Vera, J., Zuluaga, L.F.: Computing the stability number of a graph via linear and semidefinite programming. SIAM J. Optim. 18(1), 87–105 (2007) (electronic)

    MATH  MathSciNet  Google Scholar 

  40. Peng, J., Roos, C., Terlaky, T.: Self-regular functions and new search directions for linear and semidefinite optimization. Math. Program. Ser. A 93(1), 129–171 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  41. Peng, J., Roos, C., Terlaky, T.: Self-Regularity. A New Paradigm for Primal-Dual Interior-Point Algorithms. Princeton University Press, Princeton (2002)

    MATH  Google Scholar 

  42. Potra, F.A.: A quadratically convergent predictor-corrector method for solving linear programs from infeasible starting points. Math. Program. Ser. A 67(3), 383–406 (1994)

    Article  MathSciNet  Google Scholar 

  43. Potra, F.A.: An infeasible-interior-point predictor-corrector algorithm for linear programming. SIAM J. Optim. 6(1), 19–32 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  44. Potra, F.A., Sheng, R.: A superlinearly convergent primal-dual infeasible-interior-point algorithm for semidefinite programming. SIAM J. Optim. 8, 1007–1028 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  45. Roos, C.: A full-Newton step O(n) infeasible interior-point algorithm for linear optimization. SIAM J. Optim. 16(4), 1110–1136 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  46. Roos, C., Terlaky, T., Vial, J.-Ph.: Theory and Algorithms for Linear Optimization. An Interior-Point Approach. Wiley, Chichester (1997)

    MATH  Google Scholar 

  47. Sturm, J.F., Zhang, S.: Symmetric primal-dual path following algorithms for semidefinite programming. Appl. Numer. Math. 29(3), 301–315 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  48. Suzuka, A., Ryuhei, M., Yoshise, A., Tomomi, M.: Semidefinite programming based approaches to home-away assignment problems in sports scheduling. Lect. Notes Comput. Sci. 3521, 95–103 (2005)

    Article  Google Scholar 

  49. Tanabe, K.: Centered Newton method for mathematical programming. Syst. Model. Optim. 113, 197–206 (1988)

    Article  MathSciNet  Google Scholar 

  50. Tanabe, K.: Centered Newton method for linear programming: interior and ‘exterior’ point method (in Janpanese). In: Tone, K. (ed.) New Methods for Linear Programming. vol. 3, pp. 98–100. The Institute of Statistical Mathematics, Tokyo (1990)

    Google Scholar 

  51. Todd, M.J.: Semidefinite optimization. Acta Numer. 10, 515–560 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  52. Todd, M.J., Toh, K.C., Tutuncu, R.H.: On the Nesterov-Todd direction in semidefinite programming. SIAM J. Optim. 8(3), 769–796 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  53. Vandenberghe, L., Boyd, S.: A primal-dual potential reduction method for problems involving matrix inequalities. Math. Program. Ser. B 69, 205–236 (1995)

    MathSciNet  Google Scholar 

  54. Wang, G.Q., Bai, Y.Q., Roos, C.: Primal-dual interior-point algorithms for semidefinite optimization based on a simple kernel function. J. Math. Model. Algorithms 4, 409–433 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  55. Wolkowicz, H., Saigal, R., Vandenberghe, L.: Handbook of Semidefinite Programming. International Series in Operations Research & Management Science, vol. 27. Kluwer Academic, Boston (2000)

    Google Scholar 

  56. Ye, Y., Todd, M.J., Mizuno, S.: An \(O(\sqrt{n} L)\)-iteration homogeneous and self-dual linear programming algorithm. Math. Oper. Res. 19, 53–67 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  57. Zhang, Y.: On the convergence of a class of infeasible-interior-point methods for the horizontal linear complementarity problem. SIAM J. Optim. 4, 208–227 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  58. Zhang, Y.: On extending primal-dual interior-point algorithms from linear programming to semidefinite programming. SIAM J. Optim. 8(2), 365–386 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  59. Zhao, Z., Braams, B.J., Fukuda, M., Overton, M.L., Percus, J.K.: The reduced density matrix method for electronic structure calculations and role of three-index representability. J. Chem. Phys. 120, 2095–2104 (2004)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, Shahrekord University, P.O. Box 115, Shahrekord, Iran

    H. Mansouri

  2. Department of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, P.O. Box 5031, 2600, GA Delft, The Netherlands

    H. Mansouri & C. Roos

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  1. H. Mansouri
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  2. C. Roos
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Mansouri, H., Roos, C. A new full-Newton step O(n) infeasible interior-point algorithm for semidefinite optimization. Numer Algor 52, 225–255 (2009). https://doi.org/10.1007/s11075-009-9270-7

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